A line passes through #(9 ,5 )# and #(6 ,8 )#. A second line passes through #(7 ,3 )#. What is one other point that the second line may pass through if it is parallel to the first line?

Question

A line passes through #(9    ,5   )# and #(6   ,8   )#.  A second line passes through #(7    ,3  )#.  What is one other point that the second line may pass through if it is parallel to the first line?

Aaliyah Anderson · Accepted Answer

#(color(green)(0,10))#

A line passing through #(9,5)# and #(6,8)# has a slope of #color(white)("XXX")m=(Delta y)/(Delta x) = (8-5)/(6-9)=3/(-3) =-1#

All parallel lines have the same slope.

So a parallel line passing through #(7,3)# and some other point #(x,y)# will have the form: #color(white)("XXX")(y-3)/(x-7)=-1# or #color(white)("XXX")y=10-x#

Picking an arbitrary value for #x#, say #x=0# we have #color(white)("XXX")y=10# as a corresponding point.

So one point that satisfies the given requirements is #(0,10)#

Ella Armstrong · Answer

undefined To find a point through which the second line passes if it is parallel to the first line, we can use the slope of the first line. The slope of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$. Then, since the second line is parallel to the first line, it must have the same slope.

Using the points $(9, 5)$ and $(6, 8)$, we can find the slope of the first line:

$$m = \frac{8 - 5}{6 - 9} = \frac{3}{-3} = -1$$

Now, we know that the second line must also have a slope of -1 since it is parallel to the first line. We can use the slope-intercept form of a line, $y = mx + b$, to find the y-intercept ($b$) of the second line using the point $(7, 3)$. Then, we can find another point that lies on this line.

Using the point $(7, 3)$ and the slope $m = -1$, we can find the y-intercept ($b$) of the second line:

$$3 = (-1)(7) + b$$
$$b = 3 + 7$$
$$b = 10$$

Now that we have the slope ($m = -1$) and the y-intercept ($b = 10$) of the second line, we can find another point on this line. We can choose any x-value and use the equation of the line to find the corresponding y-value. Let's choose $x = 2$:

$$y = (-1)(2) + 10$$
$$y = -2 + 10$$
$$y = 8$$

So, another point through which the second line passes if it is parallel to the first line is $(2, 8)$.